Math Core

Sales tax, test scores, tips, battery life, survey results: percents are everywhere. A percent is just a ratio with a denominator of 100, so everything from the last two lessons applies. In this lesson you'll organize every basic percent question into a single equation.

Percent means "per hundred"

The word percent means "per hundred." So $35\%$ means $\dfrac{35}{100}$ , or $0.35$ . To change a percent to a decimal, divide by 100 (move the decimal point two places left).

Percent	Fraction	Decimal
$35\%$	$\dfrac{35}{100}$	$0.35$
$8\%$	$\dfrac{8}{100}$	$0.08$
$150\%$	$\dfrac{150}{100}$	$1.5$
$0.5\%$	$\dfrac{0.5}{100}$	$0.005$

A percent over $100\%$ means more than the whole. A percent under $1\%$ means less than one hundredth of the whole.

The percent equation

Every basic percent question involves three quantities: a part, a whole, and a percent. The word "of" means multiply, and "is" means equals.

The percent equation

\text{part} = \text{percent} \times \text{whole}

Write the percent as a decimal. If you know any two of the three quantities, you can solve for the third.

The three types of questions all use this one equation. Only the position of the unknown changes.

Worked example: Finding the part

What is $35\%$ of 80?

Here the whole is 80 and the percent is $0.35$ . The part is unknown.

\text{part} = 0.35 \times 80 = 28

Worked example: Finding the percent

18 is what percent of 72?

The part is 18 and the whole is 72. Let $p$ be the percent, as a decimal.

\begin{aligned} 18 &= p \times 72 \\ p &= \frac{18}{72} = 0.25 \end{aligned}

So 18 is $25\%$ of 72.

Worked example: Finding the whole

42 is $60\%$ of what number?

The part is 42 and the percent is $0.6$ . Let $w$ be the whole.

\begin{aligned} 42 &= 0.6w \\ w &= \frac{42}{0.6} = 70 \end{aligned}

Check: $0.6 \times 70 = 42$ . ✓

The proportion method

Because a percent is a ratio out of 100, you can also write a proportion:

\frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}

For "42 is $60\%$ of what number?", write $\dfrac{42}{w} = \dfrac{60}{100}$ . Cross products give $60w = 4200$ , so $w = 70$ , the same answer. Use whichever method you find clearer. The equation method is the one you'll use most in algebra.

Common mistake

A tiny percent is not the same as its decimal. $0.5\%$ is $0.005$ , not $0.5$ . So $0.5\%$ of 400 is $0.005 \times 400 = 2$ , not 200. Always divide the percent by 100, even when it's already a decimal.

Mental math with percents

Some percents are easy to find in your head, and you can build others from them.

$10\%$ : move the decimal point one place left. $10\%$ of 60 is 6.
$5\%$ : half of $10\%$ . $5\%$ of 60 is 3.
$1\%$ : move the decimal point two places left. $1\%$ of 60 is 0.6.

So $15\%$ of 60 is $6 + 3 = 9$ , and $20\%$ of 60 is $6 + 6 = 12$ .

Worked example: A restaurant tip

A meal costs $48. Find a $20\%$ tip.

$10\%$ of 48 is 4.80, so $20\%$ is $2 \times 4.80 = 9.60$ . The tip is $9.60. Check with the equation: $0.20 \times 48 = 9.6$ . ✓

Tip

Estimate first to catch mistakes. $60\%$ of something is a bit more than half of it. $125\%$ of something is more than the whole thing. If your answer doesn't fit, recheck where you put the decimal point.